how to stick an object to another object that has a wave modifier?
From your second screen it seems like wrong order of modifiers ... Solidify-Wave-Subdiv ... the order on the third one is correct (Subdiv-Wave-Solidify).
Are you sure both objects use this order?

Few Loop Cuts Ctrl+R


Same modifiers


Another way ... use the Wave modifier only for Plane and for Strip object add Surface Deform modifier with Plane as target. Be sure before Bind disable Wave modifier and enable back after binding.